State-Space Representation

Conceptual

State-Space Representation describes a dynamic system not by a single input-to-output formula, but by a state vector — the minimal set of numbers (like positions and velocities) that fully captures the system’s condition right now — plus rules for how that state changes over time. It matters because it is the shared mathematical language that Kalman filters, LQR, and MPC all speak: master it once and the rest of the modern control stack opens up.

Prerequisites

Robot Dynamics Where the A and B matrices come from — the physics of motion.

Degrees of Freedom Determines how many states a system needs.

The two equations

In words first:

State equation: how the state changes = the system’s own dynamics acting on the current state, plus how the inputs push it. As symbols: ẋ = A·x + B·u.
Output equation: what we actually measure = the parts of the state our sensors can see, plus any direct passthrough of the input. As symbols: y = C·x + D·u.

Symbol	Name	Meaning
x	state	the minimal snapshot (e.g. position, velocity)
u	input / control	what we command (e.g. motor force)
y	output / measurement	what the sensors report
A	state (system) matrix	the internal dynamics
B	input matrix	how controls enter the state
C	output matrix	which states the sensors observe
D	feedthrough matrix	direct input-to-output term (often zero)

        D
        ┌──────────────────────────┐
        │                          ▼
 u ──┬─▶[ B ]──▶(+)──▶[ ∫ ]──┬─▶[ C ]──▶(+)──▶ y
     │           ▲      x=∫ẋ │
     │           │           │
     │         [ A ]◀────────┘
     └───────────┘
   ẋ = A·x + B·u        y = C·x + D·u

Unlike a single-input single-output (SISO) PID loop tuned by hand, state-space handles multi-input multi-output (MIMO) systems naturally: A, B, C, and D are just bigger matrices, and the same math still applies.

Controllability and observability

Two questions decide whether a state-space model is usable:

Controllability — can the inputs actually drive the state anywhere we want? If a robot’s wheels can’t influence its tilt, no clever controller will fix the tilt. Controllability is the precondition for control (LQR, MPC).
Observability — can we reconstruct the full state from the measurements we have? If we only sense position but need velocity too, the state must still be inferable. Observability is the precondition for state estimation (Kalman filters).

Both are checkable by building a matrix and testing its rank: the controllability matrix [B AB A²B …] and the observability matrix [C; CA; CA² …]. Full rank means yes.

Worked example: spring-mass-damper

A mass m on a spring k with damping c. Let the state be x = [position, velocity]. Newton’s law gives:

A = [[0,    1   ],   B = [[0  ],   C = [[1, 0]],   D = [[0]]
     [-k/m, -c/m]]        [1/m]]

With m=1, k=10, c=2:

import numpy as np

m, k, c = 1, 10, 2
A = np.array([[0, 1], [-k/m, -c/m]])   # [[0, 1], [-10, -2]]
B = np.array([[0], [1/m]])             # [[0], [1]]
C = np.array([[1, 0]])                 # measure position only
D = np.array([[0]])

# Controllability: rank of [B  AB] must be 2
Co = np.hstack([B, A @ B])
print(np.linalg.matrix_rank(Co))       # 2  -> controllable

# Observability: rank of [C; CA] must be 2
Ob = np.vstack([C, C @ A])
print(np.linalg.matrix_rank(Ob))       # 2  -> observable

print(np.linalg.eigvals(A))            # -1+3j, -1-3j  -> stable

Both ranks come out 2 (full), so the system is controllable and observable. The eigenvalues -1 ± 3j have negative real parts, confirming a stable, gently oscillating decay — exactly what you’d expect from a damped spring.

State Estimation Reconstructing the full state from limited measurements.

Kalman Filter Optimal state estimation for noisy, observable systems.

LQR Optimal feedback control on a state-space model.

Model Predictive Control Optimizing over a state-space model's predicted future.

Sources

Hespanha, J. P. Linear Systems Theory, 2nd ed. Princeton University Press, 2018 — Chapters on state-space models, controllability, and observability.
Åström, K. J. & Murray, R. M. Feedback Systems: An Introduction for Scientists and Engineers, 2nd ed. Princeton University Press, 2021 — Chapter 7, “State Feedback.” Freely available at fbsbook.org.
Brunton, S. L. & Kutz, J. N. Data-Driven Science and Engineering, 2nd ed. Cambridge University Press, 2022 — Control chapter on state-space, controllability, and observability rank tests.